Nonlinear interpolation, particularly involute interpolation, in numerical control systems is highly required for the machining of gears, pump vanes, or the like. Generally, an involute curve is interpolated by a computer or an NC program generator which is separate from a numerical control system, producing linear data on a tape, and a workpiece is machined by the numerical control system using the tape,
The applicants have filed Japanese Patent Application No. 62-157302 (Japanese Laid-Open Patent Publication No. 64-2106) on an involute interpolation speed control method. According to the disclosed method, an involute curve is simply interpolated in a numerical control system, and the speed in a tangential direction is held constant irrespective of the angle.
In the proposed involute interpolation speed method, the coordinates of a point on an involute curve are given by: EQU X=R{cos (.theta.+.theta.1)+.theta. sin (.theta.+.theta.1)}+Xo EQU Y=R{sin (.theta.+.theta.1)+.theta. cos (.theta.+.theta.1)}+Yo,
where R is the radius of a base circle, and Xo and Yo are the coordinates of the center of the base circle. Then, the angle .theta. is increased in a range from .theta.=(.theta.2-.theta.1) to .theta.=(.theta.3-.theta.1) by an increment: EQU .theta.n+1=.theta.n+K/(R.multidot..theta.),
a point Xn+1, Yn+1 corresponding to the increased angle is determined according to the above equations, and the difference between the points is determined, thus interpolating the involute curve.
The speed in a tangential direction can be rendered constant by selecting the increment of .theta. to be of a value which is reduced in inverse proportion to the angle, i.e., a value of K/(R.multidot..theta.).
In the vicinity of the base circle for the involute curve, i.e., a region where the radius of curvature of the involute curve is relatively small, however, an insufficient cut or an excessive cut tends to occur due to a servo response delay, a thermal deformation of the workpiece, or the like.
FIG. 2 of the accompanying drawings shows the manner in which a workpiece is machined according to a conventional involute interpolation process. A base circle C is a circle based on which an involute curve is drawn. The base circle C has a center O represented by coordinates (Xo, Yo) and a radius R.
An involute curve In1 starts at a point Ps1 and ends at a point Pe1. An arcuate curve A1 starts at a point As1 and ends at a point Ae1. An arcuate curve A2 starts at a point As2 and ends at a point Ae2. An involute curve In2 starts at a point Ps2 and ends at a points Pe2. The positional coordinates of these points are commanded in advance by a numerical control system based on a tape or the like. Actually, another arcuate curve is interposed between the arcuate curves A1, A2 to provide a smooth junction near the point As2. However, such another arcuate curve is omitted from illustration.
A cutter W moves according to interpolation along a programmed command path which is composed of the involute curve In1, the arcuate curve A1, the arcuate curve A2, and the involute curve In2. When a workpiece is actually machined by the cutter based on the program, however, the workpiece is machined along a solid line curve Re between a point Ps3 and the point Ae1, leaving a hatched region uncut short of the commanded machining configuration, i.e., an insufficient cut.
The insufficient cut starts at the point Ps3, which is spaced from the base circle C by a radius of curvature Rs, and is progressively larger toward the point Pe1. At the point Pe1, the workpiece remains uncut by a distance De normal to the involute curve In1. After the involute curve In1 is interpolated, and when the arcuate curve A1 is interpolated, the insufficient cut is progressively reduced, converging toward the point Ae1. As can be seen from FIG. 2, the insufficient cut is large in the vicinity of the junction between the involute curve In1 and the arcuate curve A1.
Since the insufficient cut occurs in a region where high precision is required, i.e., a portion of the involute curve In1 where the radius of curvature is small and a portion of the arcuate curve A1 which is joined to the involute curve In1, the insufficient cut has posed a serious problem in machining a workpiece along an interpolated involute on a numerical control system.
The above description is directed to an insufficient cut which takes place when a workpiece is machined along a concave portion of an involute curve. However, when a workpiece is machined along a convex portion of an involute curve, an excessive cut occurs which also has posed the same problem as described above.